The NA62 experimental signature of the \(K^+\rightarrow \pi ^+\nu \bar{\nu }\)  decay is an incoming
\(K^+\) , an outgoing \(\pi ^+\) , and missing energy and momentum. The
\(K^+ \rightarrow \pi ^+ X\)  decay, where \(X\)  can be a scalar or a
pseudo-scalar particle, has the same signature. A search for a feebly
interacting particle \(X\)  gives access to physics Beyond the Standard
Model (BSM) at low energies. In a hidden-sector portal framework, the
\(X\)  particle mediates interactions between SM and hidden-sector fields
[1]}. A scalar mediator \(X\)  can mix with the SM Higgs in
inflationary [2]}, scale invariant
[3]}, and relaxion [4]} models, which all
have cosmological implications. Models where \(X\)  is an axion, which
acquires mass from the explicit breaking of the Peccei-Quinn (PQ)
symmetry [5]}, [6]}, can be a signature of the PQ mechanism
and can solve the strong CP problem [7]}, [8]}. A QCD axion with a mass of
\(\mathcal {O}(10^{-4})\)  eV could be a dark matter candidate, and
specific axion models can also solve the SM flavour problem
[9]}. In a broader class of models, \(X\)  is considered
as an axion-like particle (ALP) that acts as a pseudoscalar mediator
[10]}.
